Asymptotics of the sample coefficient of variation and the sample dispersion

The coefficient of variation and the dispersion are two examples of widely used measures of variation. We show that their applicability in practice heavily depends on the existence of sufficiently many moments of the underlying distribution. In particular, we offer a set of results that illustrate the behavior of these measures of variation when such a moment condition is not satisfied. Our analysis is based on an auxiliary statistic that is interesting in its own right. Let ( X i ) i ≥ 1 be a sequence of positive independent and identically distributed random variables with distribution function F and define for n ∈ N T n ≔ X 1 2 + X 2 2 + ⋯ + X n 2 ( X 1 + X 2 + ⋯ + X n ) 2 . Mainly using the theory of functions of regular variation, we derive weak limit theorems for the properly normalized random quantity T n , given that 1 - F is regularly varying. Following a distributional approach based on T n , we then analyze asymptotic properties of the sample coefficient of variation. As a second illustration of the same method, we then turn to the sample dispersion. We also include asymptotic properties of the first moments of these quantities. Finally, we give a distributional result on Studentʹs t-statistic which is closely related to T n . The main message of this paper is to show that the unconscientious use of some measures of variation can lead to wrong conclusions.